Having mastered the production of patterns with 3-fold rotational symmetry, 4-fold symmetries (the so-called dihedral group) can be generated in a similar way. This time, instead of adding three complex waves travelling at 120º to each other, we add four wave trains spaced at 90º (with the option of using our starting point the X axis or 45º to the X axis. Some results are show below with further work in this Gallery.
As with previous investigations, a fair amount of trial and error is required in order to produce a result that achieves a good balance between obvious symmetry and sufficient complexity to be interesting. In general, the best recipe seems to be the use of successively small amount of higher harmonics - though one can still explore considerable image variety by changing the exact amplitudes and phases of the separate harmonic contributions.
A further degree of freedom is provided by allowing some choice in the centring point from which the symmetry is generated (by pressing the "C" key). The general form of the pattern stays fixed, but the colour balances change to reflect the colours spreading from the selected central point. The Gallery shows examples where just the centre has been changed.